Highlights of the 2005 Joint Mathematics Meetings

The 2005 Joint Mathematics Meetings of the American Mathematical Society (AMS) and Mathematical Association of America (MAA) were held in Atlanta, Georgia, January 5-8. There were nearly 5,000 participants--mathematicians, exhibitors, employers, and students--who attended invited addresses, special sessions, mini-courses, the prize ceremony, exhibits, contributed paper sessions, and poster sessions. The meetings provided many opportunities for attendees to meet with colleagues old and new, at receptions, meetings and informal gatherings.

The panel discussion organized by Keith Devlin of Stanford University, showed that this longstanding conundrum is still a compelling question for today's mathematicians. The panelists were Paul Cohen of Stanford University, Donald Martin of UCLA, and Hugh Woodin of UC Berkeley. The Continuum Hypothesis (CH) is a statement about the cardinality of infinite sets. Call the cardinality of the set of integers A0 and the cardinality of the set of real numbers A1. Then CH asserts that there is no cardinal number between A0 and A1. Cohen, now 70 years old, received the Fields Medal in 1966 for showing that CH is independent of the Zermelo-Frankel axioms of set theory. CH is therefore undecidable in Zermelo-Frankel set theory. But could additional axioms be added that would allow for a proof of CH? To date, no one has come up with such axioms, but recent work of Woodin seems to move in this direction. If such axioms are found, would they be as simple, clear, and natural as the Zermelo-Frankel axioms?

In his invited address entitled "The Power and Weakness of Randomness (when you are short of time)," Avi Wigderson of the Institute for Advanced Study touched on some of the deepest and most mysterious questions in theoretical computer science today. Efficient algorithms exist to solve many real-world problems on computers. Here an efficient algorithm is one whose running time increases as a polynomial function of the size of the problem. By introducing randomness into the algorithm, one can improve the running time but one must pay the price of a small possible error in the answer. This error can be made as small as one likes. A simple example: Suppose you want to know if a polynomial expression is identically zero. The running time for simplifying the expression could be an exponential function of the size of the expression. So choose a point at random, plug it into the expression, and if you get zero, the expression is probably identically zero. You can improve the accuracy by plugging in more points. Wigderson presented a number of problems in which such probabilistic algorithms work well. He then posed a question: Are there problems for which an efficient probabilistic algorithm can be found, but for which no deterministic algorithm exists? No one knows for sure. Likewise, no one knows the answer to the famous P versus NP problem, which asks, Are there problems for which an exponential-time algorithm exists but no polynomial-time algorithm exists? One of the great achievements of theoretical computer science, Wigderson said, is the surprising fact that the answers to these two questions cannot both be "yes." If there is at least one problem that is truly hard, then efficient probabilistic algorithms, for any problem, can always be replaced by efficient deterministic ones. This achievement requires a new understanding of the age-old notion of "randomness."--- Allyn Jackson, Senior Writer and Deputy Editor of The Notices of the AMS

Hilbert's 10th problem appeared on the famous list of 23 outstanding mathematics problems proposed by David Hilbert in 1900. The 10th problem was to find a mechanical procedure for deciding the solvability of any Diophantine equation--that is, for deciding whether a polynomial equation with integer coefficients and any number of variables has a solution in integers. That no such procedure exists was proved in 1970 by Yuri Matiyasevich, using work of Julia Robinson, Martin Davis, and Hilary Putnam.

In this lecture Bjorn Poonen of UC Berkeley discussed progress on generalizations of the problem. For instance, what happens if the coefficients in the equation, and the coordinates of a solution, are allowed to be in a ring other than the ring of integers Z? For some rings the answer is obvious, for others a bit of work is required, and for still others nothing is known. One ring for which the question is unresolved is the rational numbers Q. Poonen discussed some approaches to the question, such as seeing whether the "no" answer for Z implies a "no" answer for Q. These approaches, if successful, would also disprove a conjecture of Barry Mazur about the topology of rational points on varieties. Poonen has proven that the answer is "no" for a ring that is in some sense "close to" Q, but it is unclear if the method can be generalized to Q itself.--- Allyn Jackson, Senior Writer and Deputy Editor of The Notices of the AMS

Has the Poincaré Conjecture been solved? What about the Thurston Geometrization Conjecture? These questions have been circulating around the mathematical community ever since Grigory Perelmanposted a series of remarkable papers on the web two years ago. Perelman is a mathematician in the Laboratory of Mathematical Physics in the St. Petersburg Branch of the Steklov Institute of Mathematics in Russia. As mathematicians have continued to struggle to understand his difficult and challenging papers, the jury is still out on whether the work is 100 percent correct and complete. In his lecture on Perelman's work, Bruce Kleiner of the University of Michigan shied away from giving a simple "yes" or "no" answer about whether Perelman has succeeded. But Kleiner did provide a clear flowchart for how Perelman's proof proceeds and outlined some of the main ideas involved. Essentially, Perelman has carried out a program proposed in the early 1980s by Richard Hamilton. Hamilton suggested that the Thurston Geometrization Conjecture could be solved using a tool called the Ricci flow. The presence of singularities is a major complication in Hamilton's approach. Perelman successfully implemented a surgery process that "clips off" the singular part of the manifold. Kleiner identified three key phases in Perelman's proof: 1) show that the singularities have a standard form; 2) carry out the flow-with-surgery method; and 3) study the long-time behavior of the flow with surgeries. As experts like Kleiner have worked through Perelman's proof, all the small problems that have arisen have been fixed. While these experts would probably be very surprised if the proof turns out to be wrong, no one is yet clamoring to proclaim it as correct.--- Allyn Jackson, Senior Writer and Deputy Editor of The Notices of the AMS

The gorgeous, full-color graphics painted a picture of outer space as filled with "tubes" that function as superhighways weaving around the solar system. Jerrold Marsden of Caltech took his audience on a cosmic journey in his lecture "New Methods in Celestial Mechanics and Mission Design," which reported on work by a group of researchers in the United States and Germany. The central mathematical ideas go back to the great nineteenth century mathematician Henri Poincaré, who did monumental work in celestial mechanics. In particular he studied the three-body problem, a dynamical system in which three massive bodies move under mutual gravitational attraction. In such a system, the tangle of gravitational forces creates tubular passageways in the space between the bodies; these passageways are realized geometrically as invariant manifolds of the dynamical system.

The entire solar system is threaded with these tubes, Marsden explained, and they can be used to transport spacecraft from one planet to another with very little energy. What makes this method so efficient is the use of the unstable orbits of the dynamical system, which Marsden noted are actually very easy for the spacecraft to travel on. (Marsden confessed that he never understood why in the field of dynamical systems so much attention is paid to the stable orbits, while the unstable orbits are so useful and interesting.) These interplanetary transport tubes have been used in previous space missions, such as the Genesis craft that collected solar wind data, and will be used in the Jupiter Icy Moons orbit, to be launched a decade from now.

Marsden's lecture was presented in the AMS Special Session on Current Events, organized by David Eisenbud, AMS president and director of the Mathematical Sciences Research Institute in Berkeley. Talks in the session and speakers are below:

The Green-Tao Theorem on Primes in Arithmetic Progression: A Dynamical Point of View, Bryna Kra

The Prizes and Awards booklet (pdf) includes all the AMS, AMS-MAA-SIAM, AWM, JPBM and MAA prizes presented at the 2005 Joint Mathematics Meetings, with citations, as well as brief biographies of and responses from each winner. Shown here is Barry Cipra (right), who received the Communications Award from the Joint Policy Board for Mathematics.--- Annette Emerson, AMS Public Awareness Officer

How fast should Lance Armstrong pedal during the downhill stretches of the Tour de France? Michael Scott Gordon presented his answer in a lecture that combined calculus and physics. The power generated by a cyclist traveling at a constant velocity is a function of the drag coefficient, the mass of the rider and the cycle, friction, and the uphill grade. This can be expressed as an equation, the dominating term of which depends on whether the cyclist is traveling uphill or downhill. Thus, it is reasonable to ask whether it is possible to minimize the amount of work required to complete a course in a given amount of time.

The AMS exhibit included books, Mathematical Reviews® and MathSciNet, an online connection to the AMS Bookstore, a meeting place for AMS authors, and the AMS Membership booth, at which people could learn more about the Society and pick up giveaways such as calendars, the Mathematics Awareness Month poster, postcards, and various materials on AMS programs. The Society also hosted a booth for the Mathematics Genealogy Project.

The AMS also hosted a reception for Mathematical Reviews® editors and reviewers, and another for AMS authors. Shown below at the Mathematical Reviews® reception are (left to right) Jane Kister (past Executive Editor of Mathematical Reviews®), David Eisenbud (AMS President) and Carol Hutchins (Head Librarian at the Courant Institute of Mathematical Sciences Library at New York University).

Although the AMS and MAA co-sponsored the joint meetings, several other organizations also held meetings, sponsored prizes, and hosted sessions: the Association for Symbolic Logic, the Association for Women in Mathematics, the National Association of Mathematicians, the National Science Foundation, Pi Mu Epsilon, the Rocky Mountain Mathematics Consortium, the Society for Industrial and Applied Mathematics, and the Young Mathematicians Network. See the complete program.